Feng Gang National Laboratory of Communication,UESTC Aug 2017 Ver 1.4 Queuing model for Birth-Death Process(cont'd) Some special Birth-death Processes Poisson Process:=0 for all n≥0 n=元for all n≥0 The probability of k arrivals in T: p(k)=(AT)e-i/k! Inter-arrival time:Exponential distribution: f(x)=le x≥0 -Queuing system M/M/1：元，=z for alln≥0 =u for alln≥0 Number of customers in the system geometric distribution: p(k)=(1-p)p The average waiting time from arrival until the end of service for FIFO discipline 1-p (p=元1） 2616009:Network Traffic Engineering 2:Call-level Models and Admission Control Page.6
2616009: Network Traffic Engineering Feng Gang National Laboratory of Communication, UESTC Aug 2017 Ver 1.4 2: Call-level Models and Admission Control Page.6 Queuing model for Birth-Death Process(cont’d) • Some special Birth-death Processes - Poisson Process : The probability of k arrivals in T: Inter-arrival time: Exponential distribution: - Queuing system M/M/1 : 0 for all n 0 n n for all n 0 for all n 0 n for all n 0 n p(k) ( T) e / k! k T ( ) 0 f x e x x Number of customers in the system : geometric distribution: k p(k) (1 ) The average waiting time from arrival until the end of service for FIFO discipline 1 1/ T ( / )
Feng Gang National Laboratory of Communication,UESTC Aug 2017 Ver 1.4 Queuing model for Birth-Death Process(cont'd) The Erlang B Loss Formula When the number of sources is assumed to be infinite,with a total aggregate traffic load of A Erlangs and with assumption that calls which arrive when all C servers are busy are cleared from the system and do not return.The Erlang B loss formula is: A/C! B(C,A)= 41m (A=元10) 1=0 The Erlang B distribution for the number of busy servers is A"n! 三术因 2616009:Network Traffic Engineering 2:Call-level Models and Admission Control Page.7
2616009: Network Traffic Engineering Feng Gang National Laboratory of Communication, UESTC Aug 2017 Ver 1.4 2: Call-level Models and Admission Control Page.7 Queuing model for Birth-Death Process(cont’d) • The Erlang B Loss Formula - When the number of sources is assumed to be infinite, with a total aggregate traffic load of A Erlangs and with assumption that calls which arrive when all C servers are busy are cleared from the system and do not return. The Erlang B loss formula is: - The Erlang B distribution for the number of busy servers is C n n C A n A C B C A 0 / ! / ! ( , ) C k k n n A k A n P 0 / ! / ! (A / )
Feng Gang National Laboratory of Communication,UESTC Aug 2017 Ver 1.4 Queuing model for Birth-Death Process(cont'd) The Erlang-Engset Loss Formula If the number of source is finite while blocked calls are cleared The Probability of finding X servers busy is given by the Erlang- Engest distribution.Letting Sx denote the probability of finding X servers busy when there are S sources and the offered traffic per idle source is b Erlangs S: 2616009:Network Traffic Engineering 2:Call-level Models and Admission Control Page.8
2616009: Network Traffic Engineering Feng Gang National Laboratory of Communication, UESTC Aug 2017 Ver 1.4 2: Call-level Models and Admission Control Page.8 Queuing model for Birth-Death Process(cont’d) • The Erlang-Engset Loss Formula - If the number of source is finite while blocked calls are cleared . The Probability of finding X servers busy is given by the ErlangEngest distribution. Letting Sx denote the probability of finding X servers busy when there are S sources and the offered traffic per idle source is b Erlangs C i i x b i S X S S 0
Feng Gang National Laboratory of Communication,UESTC Aug 2017 Ver 1.4 Multiservice Loss Systems A loss system is a collection of resources to which calls,each with an associated holding time and class,arrive at random instances. 。 An arriving call either is admitted into the system or is blocked and lost 。 The admittance decision is based on the call's class and the system's state C bandwidth units=C servers ,41,b Collection of calls departure Resources :arrival rate System state n (Carried traffic) :service rate b,bandwidth class j Lost (blocked or overflow traffic streams) 2616009:Network Traffic Engineering 2:Call-level Models and Admission Control Page.9
2616009: Network Traffic Engineering Feng Gang National Laboratory of Communication, UESTC Aug 2017 Ver 1.4 2: Call-level Models and Admission Control Page.9 Multiservice Loss Systems • A loss system is a collection of resources to which calls, each with an associated holding time and class, arrive at random instances. • An arriving call either is admitted into the system or is blocked and lost • The admittance decision is based on the call’s class and the system’s state Collection of Resources System state n calls Lost (blocked or overflow traffic streams) departure j j j , , b : arrivalrate j :service rate j : bandwidth class j j b C bandwidth units=C servers (Carried traffic)
Feng Gang National Laboratory of Communication,UESTC Aug 2017 Ver 1.4 Loss Networks with Fixed Routing PBX Central Example:A telephone network with star topology switch 6 classes System state:n=(n,...n),n:class k in progress PBX PBX -s:the set for all state -k:the set of state with room for another class-k call O PBX neS if and only if n+ees Where ex is the vector of all zeros except for a one in the kth component Clocking Probability of a class-k call B=1- G where -20器 and nes k=1 neSx 1=1 n! Pr=h/ue 2616009:Network Traffic Engineering 2:Call-level Models and Admission Control Page.10
2616009: Network Traffic Engineering Feng Gang National Laboratory of Communication, UESTC Aug 2017 Ver 1.4 2: Call-level Models and Admission Control Page.10 Loss Networks with Fixed Routing • Example: A telephone network with star topology - 6 classes - System state: , : class k in progress - S : the set for all state - Sk: the set of state with room for another class-k call - if and only if Where is the vector of all zeros except for a one in the kth component • Clocking Probability of a class-k call PBX PBX PBX PBX Central switch ( ,... ) n n1 n6 nk k nS S k n e k e G G B k k 1 where n S 1 6 ! k k n k n ρ G k and k l l l n l k n ρ G n S 1 6 ! k k k /